$\begingroup$Why do you go through the stage of the integration? Why not merely ask (for your first question) how to express $a\ x^{3.5}$ in terms of MeijerG?$\endgroup$
– David G. StorkJan 10 '16 at 3:09

$\begingroup$Unfortunately, there's no argument that available in "MeijerG" to represent the integrand in MeijerG form.$\endgroup$
– làntèrnJan 10 '16 at 3:18

$\begingroup$I do not know of a direct Mathematica conversion routine for this. Might get some help from e.g. 1, 2, 3.$\endgroup$
– Daniel LichtblauJan 10 '16 at 16:59

There is an undocumented(!) function that essentially performs a Mellin transform through lookup. In your case, however, only one of your four examples actually has a sensible Mellin transform (and thus, a Meijer $G$ representation):

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